Denominators in Lusztig's asymptotic Hecke algebra via the Plancherel formula

Abstract

Let W be an extended affine Weyl group, H be the affine Hecke algebra of the corresponding affine Weyl group over the ring C[q, q −1 ], and J be Lusztig's asymptotic Hecke algebra, viewed as a based ring with basis {tw}. Viewing J as a subalgebra of the (q −1 )-adic completion of H via Lusztig's map φ, we use Harish-Chandra's Plancherel formula for p-adic groups to show that the coefficient of Tx in tw is a rational function of q, depending only on the two-sided cell containing w, with no poles outside of a finite set of roots of unity that depends only on W . In type Ãn and type C2, we show that the denominators all divide a power of the Poincaré polynomial of the finite Weyl group. As an application, we conjecture that these denominators encode more detailed information bout the failure of the Kazhdan-Lusztig classification at roots of the Poincaré polynomial than is currently known.

On Lusztig’s asymptotic Hecke algebra for 𝑆𝐿₂

“…This generalizes a similar statement for the functions ν → trace (π ν , t w ) proven in [Daw23] and confirms an expectation of Braverman and Kazhdan.…”

“…In this section, we show that matrix coefficients with respect to a basis defined via the idempotents t d,ρ depend in fact algebraically on the unramified character ν. The proof is similar to proof of regularity of the function trace (π , t w ) established in [Daw23].…”

“…where E J is a subring of C(G) defined spectrally via the Paley-Wiener theorem. In [Daw23], we proved injectivity of η. In the present paper, we prove surjectivity for all but four cells occurring only in types E n .…”

“…At several points we will want to specialize q to some q > 1 so as to use the tools of harmonic analysis. If q is the cardinality of the residue field of F , then H| q=q is the Iwahori-Hecke algebra of G, and J will be a subalgebra of the Harish-Chandra Schwartz algebra, as shown in [BK18] and [Daw23].…”

A coherent categorification of the based ring of the lowest two-sided cell

“…In [BK18], Braverman and Kazhdan gave an interpretation of J in terms of harmonic analysis, casting J as an algebraic version of C(G ∨ ) I (they also defined a ring J doing the same for the full algebra C(G ∨ )) by defining an map J → C(G ∨ ) I . In [Daw21], the author showed that this morphism was essentially the specialization of φ −1 for q = q, and in particular was an injection.…”